Review

Question 1

E(X)

Answer 1

Nb. E(X) = μ_x

Question 2

E{g(X)}

Answer 2

e.g. g(X) = x²

Question 3

The three expected value rules

Answer 3

• E(X+Y+Z) = E(X) + E(Y) + E(Z)
• E(bX) = bE(X)
• E(b) = b

Question 4

Two formulae for population vairance of a discrete random variable, σ² using

Answer 4

• var(X) = σ² = E {(X - μ_x)²}
• var(X) = σ² ​ = E (X²) - μ_x²

Question 5

cov (X,Y)

Answer 5

Question 6

If X and Y are independent

then E[g(X)h(Y)] =

Answer 6

E[g(X)h(Y)] = E[g(X)] E[h(Y)]

Nb. This means that if X and Y are independent then E(XY) = E(X) E(Y)

Question 7

If X is a random variable with X = μ_x + u where u is a pure random component, what is the expectation and variance of u?

Answer 7

E(u)=0 and σ_u² = σ_X² = E(u²)​

Question 8

Covariance of X and Y

Answer 8

Cov(X,Y)= σ_XY = E{(X-μ_X)(Y-μ_Y)}

Question 9

1) If Y = V+W, Cov (X,Y) =
2) If Y=bZ, Cov (X,Y) =
3) If Y = V + b, Cov (X,Y)=

Answer 9

1) If Y = V+W, Cov (X,Y) = Cov(X,V) + Cov (X,W)
2) If Y=bZ, Cov (X,Y) = bCov(X,Z)
3) If Y = V + b, Cov (X,Y) = 0

Question 10

Formula for the sample covariance of X and Y

Answer 10

s_{XY =} Σ(x_i-x̄)(yi-ȳ)/(n-1)

Question 11

If Y = V + W, var(Y) =

If Y=bZ, then var(Y) =

If Y=b, then var(Y) =

Answer 11

If Y = V + W, var(Y) = var(V) + var(W) +2cov(X,Y)

If Y=bZ, then var(Y) = b²var(Z)

If Y=b, then var(Y) = 0

Question 12

Prove the covariance of X and Y is zero if X and Y are independent variables

Answer 12

If X and Y are independent,

Cov(X,Y) = E{(X-μ_x)(Y-μ_Y)} = E(X-μ_x)E(Y-μ_Y)

= (E(X)-μ_x)(E(Y)-μ_Y)

= (μ_x-μ_x)(μ_Y-μ_Y) = 0

Question 13

Population correlation coefficient

Answer 13

ρ_XY = σ_XY / √(σ_X² + σ²_Y)

Question 14

E(x̄) =

σ_x̄² =

Answer 14

var(x̄)=σ_X²/n

Question 15

Bias is

Answer 15

Bias is the difference between an estimator’s expected value and the value of the population characteristic

Question 16

One estimator is more efficient than another if

Answer 16

One estimator is more efficient than another if it is more likely to give an accurate estimate, i.e. its probability density function is more concentrated around the true value - it has the smaller variance out of two unbiased estimators

Question 17

Mean square error of an estimator Z of parameter θ

• two definitions

Answer 17

MSE (Z) = E{(Z-θ)²}

MSE (Z) = Var(Z) + bias²